# Confusion about the definition of reduced scheme

I'm following Eisenbud-Harris Geometry of Schemes, and I am a little bit confused about how they define the notion of a reduced scheme. In the affine case, if $$X = \text{Spec} \, R$$, then setting $$X_{\mathrm{red}} = \text{Spec}\, R_{\mathrm{red}}$$ where $$R_{\mathrm{red}}$$ is the ring $$R$$ modulo its nil radical, we say that $$X$$ is reduced if $$X = X_{\mathrm{red}}$$.

For an arbitrary scheme $$X$$, they define a quasi-coherent sheaf called the nilradical, that assigns to each open subset $$U$$ of $$X$$, the nilradical of $$\mathcal{O}_X(U)$$, being $$\mathcal{O}_X$$ the structure sheaf of $$X$$, and then say that the scheme $$X$$ is reduced if its associated closed subscheme $$X_{\mathrm{red}}$$ is equal to $$X$$.

What does the statement "its associated closed subscheme $$X_{\mathrm{red}}$$" mean?

The definition Eisenbud and Harris give of a closed subscheme $$Y$$ of a scheme $$X$$ is a closed topological subspace $$|Y| \subset |X|$$ with a sheaf $$\mathcal{O}_Y$$ that is the quotient sheaf of the structure sheaf of $$X$$ by a quasi-coherent sheaf of ideals $$\mathcal{J}$$ such that the intersection of $$Y$$ with any affine open subset $$U \subset X$$ is the closed subschema associated to the ideal $$\mathcal{J}(U)$$.

Since we have a quasi-coherent sheaf $$\mathcal{N}$$, we can quotient the structure sheaf $$\mathcal{O}_X$$ of $$X$$ by $$\mathcal{N}$$. Is then
$$X_{\mathrm{red}} = (X, \mathcal{O}_X/\mathcal{N})\, ?$$ The problem is that the definition of a closed subscheme $$Y$$ of a scheme $$X$$ says that the topological space $$|Y|$$ has to be a closed subspace of the topological space $$|X|$$, but in this case it would be the same space and not a proper subspace.

• Indeed, in this case the two schemes have the same underlying topological space. This is not a problem, however. – Sasha Nov 16 '18 at 15:40